Let $S$ Schwartz space on $\mathbb{R}^2$. For $f\in S$, let $T(f):=\int _{-\infty}^{\infty} f(0,y)dy$ and $T_{\theta} (f):=T(f(x \cos{\theta} -y \sin{\theta}, x\sin{\theta} +y\cos{\theta}))$ and we note $F$ Fourier transformation. Then $F(\{T_\theta\}_{\theta \in \mathbb{R}})=\{T_\theta\}_{\theta \in \mathbb{R}}$.
I proved $T_{\theta} \in S'$ by calculation, so I want to prove $F:\{T_\theta\} \to \{T_\theta\}$ is surjective. But I can't prove.